Probably my Western sensibilities, but I have no idea how busy Istiklal Street is and if the shops are open on a Sunday walk.
I note that it's not much discussed in Rick's Istanbul book, but I'm counting on these references: p. 46 says that Istiklal Street is "always open." That's different from the listings on the same page which specify that the Grand Bazaar is closed on Sundays and the Spice Market has slightly shorter hours on Sundays. (And on p. 366, the Turkish Handicrafts Center is closed Mondays.)
Yes, the shops in Istiklal Avenue are 7-days open and the avenue is always crowded on Sundays.
By the way, there is no such thing like New District here in Istanbul. The area where Istiklal Avenue is called as 'Beyoğlu'.